Asymptotics for the reciprocal and shifted quotient of the partition function
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910725423759360 |
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| author | Banerjee, Koustav Paule, Peter Radu, Cristian-Silviu Schneider, Carsten |
| author_facet | Banerjee, Koustav Paule, Peter Radu, Cristian-Silviu Schneider, Carsten |
| contents | Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $k\in \mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics for the reciprocal and shifted quotient of the partition function Banerjee, Koustav Paule, Peter Radu, Cristian-Silviu Schneider, Carsten Number Theory Symbolic Computation Combinatorics 05A16, 05A20, 11P82 Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $k\in \mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest. |
| title | Asymptotics for the reciprocal and shifted quotient of the partition function |
| topic | Number Theory Symbolic Computation Combinatorics 05A16, 05A20, 11P82 |
| url | https://arxiv.org/abs/2412.02257 |